Methods Summary

In real-time, dividing deaths-to-date by cases-to-date leads to a biased estimate of the case fatality ratio (CFR), because this calculation does not account for delays from confirmation of a case to death, and under-reporting of cases.
Using the distribution of the delay from hospitalisation-to-death for cases that are fatal, we can estimate how many cases so far are expected to have known outcomes (i.e. death or recovery), and hence adjust the naive estimates of CFR to account for these delays.
The adjusted CFR does not account for under-reporting. However, the best available estimates of CFR (adjusting or controlling for under-reporting) are in the 1% - 1.5% range [1–4]. Large studies in China and South Korea estimating the CFR at 1.38% (95% CrI: 1.23–1.53%)[2] and 1.4% (95% CrI: 1.2-1.7%)[4] respectively. Based on these studies, and for simplicity, we assume a baseline CFR of 1.4% for our analysis.
If a country has an adjusted CFR that is higher (e.g. 20%), it suggests that only a fraction of cases have been reported (in this case, 1.4/20=7.0% cases reported approximately).

Current estimates for percentage of symptomatic cases reported for countries with greater than ten deaths

Temporal variation

Figure 1: Temporal variation in reporting rate. We calculate the percentage of symptomatic cases reported on each day a country has had more than ten deaths. We then fit a Generalised Additive Model to these data (see Temporal variation model fitting section for details), highlighting the temporal trend of each countries reporting rate. The shaded region is the 95% CI of fitted GAM.

Current estimates

Figure 2: Plotting the estimates for the proportion of symptomatic cases reported in different countries using cCFR estimates. Blue shading is the 2.5% - 97.5% confidence range.

Table of current estimates

Country	Percentage of symptomatic cases reported (95% CI)	Total cases	Total deaths
Afghanistan	18% (12% - 44%)	1949	60
Albania	21% (13% - 50%)	766	31
Algeria	6.3% (5% - 12%)	3848	444
Andorra	18% (12% - 39%)	753	42
Argentina	15% (11% - 29%)	4272	214
Armenia	50% (30% - 100%)	1932	30
Australia	84% (59% - 100%)	6746	90
Austria	29% (23% - 46%)	15364	580
Azerbaijan	63% (36% - 100%)	1766	23
Bahamas	6% (3.2% - 20%)	80	11
Bangladesh	18% (13% - 45%)	7103	163
Belarus	73% (50% - 100%)	13181	84
Belgium	5.4% (4.6% - 9.3%)	47859	7501
Bolivia	11% (7.2% - 26%)	1110	59
Bosnia and Herzegovina	22% (15% - 47%)	1588	62
Brazil	7.6% (6.4% - 15%)	78162	5466
Bulgaria	16% (11% - 36%)	1447	64
Burkina Faso	14% (9.3% - 31%)	641	43
Cameroon	20% (13% - 45%)	1832	61
Canada	12% (10% - 23%)	51587	2996
Chile	50% (37% - 100%)	14885	216
China	24% (20% - 32%)	83944	4637
Colombia	15% (11% - 30%)	6211	278
Cote dIvoire	63% (32% - 100%)	1238	14
Croatia	30% (20% - 61%)	2062	67
Cuba	18% (12% - 43%)	1467	58
Cyprus	40% (23% - 100%)	843	20
Czechia	33% (25% - 59%)	7579	227
Democratic Republic of the Congo	12% (7.2% - 30%)	500	31
Denmark	18% (14% - 32%)	9008	443
Dominican Republic	16% (13% - 33%)	6652	293
Ecuador	13% (10% - 26%)	24675	883
Egypt	8.8% (6.9% - 18%)	5268	380
Estonia	33% (22% - 69%)	1666	50
Finland	20% (15% - 38%)	4906	206
France	5.1% (4.3% - 8.2%)	128442	24087
Germany	25% (21% - 40%)	159119	6288
Ghana	62% (33% - 100%)	1671	16
Greece	19% (14% - 34%)	2576	139
Guatemala	20% (11% - 67%)	585	16
Guernsey	19% (9.9% - 56%)	251	13
Honduras	7.6% (5.3% - 16%)	771	71
Hungary	6.8% (5.3% - 13%)	2775	312
India	17% (14% - 35%)	33050	1074
Indonesia	8.7% (7% - 17%)	9771	784
Iran	15% (13% - 24%)	93657	5957
Iraq	19% (13% - 37%)	2003	92
Ireland	13% (11% - 24%)	20253	1190
Isle of Man	14% (8.1% - 37%)	313	21
Israel	68% (51% - 100%)	15834	215
Italy	7.3% (6.2% - 11%)	203591	27682
Japan	26% (21% - 51%)	14088	415
Jersey	12% (7.2% - 32%)	286	21
Kazakhstan	77% (45% - 100%)	3205	25
Kenya	19% (10% - 59%)	384	15
Kosovo	26% (15% - 74%)	799	22
Kuwait	87% (50% - 100%)	3740	24
Latvia	54% (28% - 100%)	849	15
Lebanon	32% (19% - 74%)	721	24
Liberia	5.5% (3.2% - 18%)	141	16
Lithuania	31% (20% - 67%)	1375	45
Luxembourg	45% (31% - 84%)	3769	89
Malaysia	59% (42% - 100%)	5945	100
Mali	9.7% (5.9% - 29%)	482	25
Mexico	5.4% (4.4% - 11%)	17799	1732
Moldova	23% (17% - 51%)	3771	111
Morocco	18% (13% - 38%)	4321	168
Netherlands	7.4% (6.1% - 12%)	38802	4711
New Zealand	65% (36% - 100%)	1129	19
Niger	20% (12% - 48%)	713	32
Nigeria	15% (9.9% - 40%)	1728	51
North Macedonia	17% (11% - 36%)	1442	73
Norway	40% (30% - 69%)	7667	202
Pakistan	27% (21% - 56%)	15759	346
Panama	27% (20% - 54%)	6378	178
Peru	18% (14% - 38%)	33931	943
Philippines	12% (9.4% - 22%)	8212	558
Poland	15% (12% - 29%)	12640	624
Portugal	22% (18% - 39%)	24505	973
Puerto Rico	14% (9.6% - 29%)	1433	86
Romania	14% (11% - 25%)	11978	675
Russia	48% (39% - 100%)	99399	972
San Marino	12% (8% - 27%)	563	41
Saudi Arabia	68% (50% - 100%)	21402	157
Serbia	36% (27% - 76%)	8724	173
Singapore	100% (100% - 100%)	15641	14
Sint Maarten	5.1% (2.9% - 16%)	76	13
Slovakia	54% (31% - 100%)	1391	22
Slovenia	17% (12% - 31%)	1418	89
Somalia	7.7% (4.8% - 26%)	582	28
South Africa	35% (25% - 74%)	5350	103
South Korea	55% (42% - 83%)	10765	247
Spain	8.5% (7.2% - 14%)	212917	24275
Sudan	3.9% (2.5% - 13%)	375	28
Sweden	6.3% (5.2% - 11%)	20302	2462
Switzerland	22% (18% - 35%)	29324	1407
Thailand	58% (38% - 100%)	2954	54
Tunisia	23% (15% - 51%)	980	40
Turkey	28% (24% - 53%)	117589	3081
Ukraine	24% (18% - 53%)	9866	250
United Arab Emirates	74% (52% - 100%)	11929	98
United Kingdom	4.8% (4% - 8.4%)	165221	26097
United Republic of Tanzania	12% (6.4% - 42%)	480	16
United States of America	13% (11% - 23%)	1039909	60966
Uruguay	40% (21% - 100%)	630	15

Table 1: Estimates for the proportion of symptomatic cases reported in different countries using cCFR estimates based on case and death timeseries data from the ECDC. Total cases and deaths in each country is also shown. Confidence intervals calculated using an exact binomial test with 95% significance.

Adjusting for outcome delay in CFR estimates

During an outbreak, the naive CFR (nCFR), i.e. the ratio of reported deaths date to reported cases to date, will underestimate the true CFR because the outcome (recovery or death) is not known for all cases [5]. We can therefore estimate the true denominator for the CFR (i.e. the number of cases with known outcomes) by accounting for the delay from confirmation-to-death [1].

We assumed the delay from confirmation-to-death followed the same distribution as estimated hospitalisation-to-death, based on data from the COVID-19 outbreak in Wuhan, China, between the 17th December 2019 and the 22th January 2020, accounting right-censoring in the data as a result of as-yet-unknown disease outcomes (Figure 1, panels A and B in [7]). The distribution used is a Lognormal fit, has a mean delay of 13 days and a standard deviation of 12.7 days [7].

To correct the CFR, we use the case and death incidence data to estimate the proportion of cases with known outcomes [1,6]:

\[ u_{t} = \frac{ \sum_{j = 0}^{t} c_{t-j} f_j}{c_t}, \]

where \(u_t\) represents the underestimation of the proportion of cases with known outcomes [1,5,6] and is used to scale the value of the cumulative number of cases in the denominator in the calculation of the cCFR, \(c_{t}\) is the daily case incidence at time, \(t\) and \(f_t\) is the proportion of cases with delay of \(t\) between confirmation and death.

Approximating the proportion of symptomatic cases reported

At this stage, raw estimates of the CFR of COVID-19 correcting for delay to outcome, but not under-reporting, have been calculated. These estimates range between 1% and 1.5% [1–3]. We assume a CFR of 1.4% (95% CrI: 1.2-1.7%), taken from a recent large study [3], as a baseline CFR. We use it to approximate the potential level of under-reporting in each country. Specifically, we perform the calculation \(\frac{1.4\%}{\text{cCFR}}\) of each country to estimate an approximate fraction of cases reported.

Temporal variation model fitting

We estimate the level of under-reporting on every day for each country that has had more than ten deaths. We then fit a Generalised Additive Model (GAM) of the form \[ \mathbb{E}[\log(D)] = \beta_0 + \beta_1 x_1 + ... + \beta_p x_p,\] specifying a Poisson distribution on deaths (D) as the response variable. The model has a log-link function and a log-offset (\(\kappa\)) consisting of the daily known-outcomes \(u_t\) and the cCFR estimate for that country on that day \(\text{cfr}_t\). The model can then be written as \[ D \sim s(t) + \underbrace{\log(u_t c_t) + \log(\text{cfr}_t)}_{:=log(κ)} \] where \(s(t)\) is a smoothing spline, fitted through the time points (days) for which we have data.

Limitations

Implicit in assuming that the under-reporting is \(\frac{1.4\%}{\text{cCFR}}\) for a given country is that the deviation away from the assumed 1.4% CFR is entirely down to under-reporting. In reality, burden on healthcare system is a likely contributing factor to higher than 1.4% CFR estimates, along with many other country specific factors.

The following is a list of the other prominent assumptions made in our analysis:

We assume that people get tested upon hospitalisation. A few examples where this is not the case are Germany and South Korea, where people can get tested earlier.
We assume that hospitalisation to death from early Wuhan is representative of the all the other countries (by using the distribution parameterised using early Wuhan data) and that all countries have the same risk and age profile as Wuhan.
Severity of COVID-19 is known to increase with age. Therefore, countries with older populations will naturally see higher death rates. We are extending this analysis to adjust for the age distribution for countries with more than five reported deaths and where age distribution data is available.
All results are linked and biased by the baseline CFR, assumed at 1.4% [3].
The under-reporting estimate is sensitive to the baseline CFR, meaning that small errors in it lead to large errors in the estimate for under-reporting.
There are several sources of uncertainty in this analysis: the reported mean and SD of the delay distribution; the cCFR for each country (on each day) and the baseline CFR. We use the lower and upper mean and SD of the delay distribution to produce the widest interval. The other two uncertainties are just carried through directly in the subsequent calculations, which is crude. An ongoing extension is developing this into a fully Bayesian model, which deals with these different uncertainties more rigorously.

Code and data availability

The code is publically available at https://github.com/thimotei/CFR_calculation. The data required for this analysis is a time-series for both cases and deaths, along with the corresponding delay distribution. We scrape this data from ECDC, using the NCoVUtils package [8].

References

1 Russell TW, Hellewell J, Jarvis CI et al. Estimating the infection and case fatality ratio for covid-19 using age-adjusted data from the outbreak on the diamond princess cruise ship. medRxiv 2020.

2 Verity R, Okell LC, Dorigatti I et al. Estimates of the severity of covid-19 disease. medRxiv 2020.

3 Guan W-j, Ni Z-y, Hu Y et al. Clinical characteristics of coronavirus disease 2019 in china. New England Journal of Medicine 2020.

4 Shim E, Mizumoto K, Choi W et al. Estimating the risk of covid-19 death during the course of the outbreak in korea, february-march, 2020. medRxiv 2020.

5 Kucharski AJ, Edmunds WJ. Case fatality rate for ebola virus disease in west africa. The Lancet 2014;384:1260.

6 Nishiura H, Klinkenberg D, Roberts M et al. Early epidemiological assessment of the virulence of emerging infectious diseases: A case study of an influenza pandemic. PLoS One 2009;4.

7 Linton NM, Kobayashi T, Yang Y et al. Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: A statistical analysis of publicly available case data. Journal of Clinical Medicine 2020;9:538.

8 Abbott S MJ Hellewell J. NCoVUtils: Utility functions for the 2019-ncov outbreak. doi:105281/zenodo3635417 2020.

Aim